If G is a Lie group whose adjoint representation preserves a nondegenerate symmetric bilinear form on its Lie algebra (e.g. a semisimple group) and F is the fundamental group of a closed oriented surface S, then the spaces of equivalence classes of representations F-> G (equivalently gauge-equivalence classes of flat G-connections over S) enjoys a rich symplectic geometry. These symplectic manifolds generalize the Kahler structures on the Jacobi variety, moduli of holomorphic vector bundles, and Teichmueller space. This talk will describe this geometry and several open questions about these symplectic manifolds.